We theoretically study the properties of the focal region of abruptly autofocusing and, their variant, autodefocusing ring-Airy beams, under the action of a conical phase gradient. By expanding the analysis of 1D Airy beams to cylindrically symmetric Airy beams, we derive analytic formulas for the position and dimensions of the focus. Our analysis covers in a unified way both beam types, and numerical simulations over a broad parameters range are in excellent agreement with our theoretical predictions. Our results allow the tailoring of the focal region both in size and in position by tuning the initial parameters.
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Abruptly autofocusing beams, often referred as CABs or ring-Airy beams, are a subject of intense research effort [1–8], since they were first introduced [9, 10]. Their most exciting property is that instead of spreading due to diffraction, they abruptly autofocus [9,10], exhibiting a more than two orders of magnitude increase of their intensity. Likewise, in the non-linear propagation regime , these beams preserve their abrupt autofocusing, with a minimal nonlinear focal shift. This makes them ideal candidates for applications that require controlled energy deposition, with minimal effects before the focus . For example, ring-Airy beams are advantageous compared to Gaussian or Bessel beams in multiphoton polymerization  and in two color laser plasma THz generation . Furthermore, a tighter focus, and thus further increase of the focal intensity, can be achieved by using a lens. As it has been recently shown , due to the twin wave nature of ring-Airy beams, focusing by a lens leads to two foci instead of one. This counterintuitive behavior is characteristic for a larger family of waves, referred to as Janus waves . Although this effect is interesting and beneficial to some applications , it prevents the use of focusing elements in applications where the twin focus is undesirable. An alternative approach for controlling the focal properties is to use a conical phase gradient. As it has been recently shown  for the case of abruptly autofocusing beams this conical phase acts in a cooperative action with the autofocusing properties of the beam and provides some additional control to the focus position.
On the other hand, besides the extensive work in the field, to our knowledge analytical expressions that predict the dimensions and position of the focus for abruptly autofocusing and autodefocusing beams are sparse and incomplete. The position of the focus is rather accurately predicted for autofocusing beams [3, 11] even in the case of the presence of a conical phase gradient , but to our knowledge no analytical description exists for their autodefocusing variant . Furthermore, analytical formulas that predict the focal dimensions  are applicable only to abruptly autofocusing beams for a limited range of beam parameters, and without the presence of a conical phase gradient.
In this work, we follow a unified theoretical analysis for both autofocusing, and autodefocusing, ring-Airy beams in the presence of a conical phase gradient. Starting from the analytical expressions that describe the propagation of 1D Airy beams , we derive generic analytical expressions that predict the position and dimensions of the focus for both types of beams. Our theoretical predictions are in excellent agreement with numerical simulations, performed over an extended parameters range.
2. Radially symmetric Airy beams
The 1D Airy beams impart their unique properties to ring-Airy beams, their rotationally symmetric variant. As shown in Fig. 1(a) the intensity profile of a ring-Airy beam  is comprised by concentric rings that become thinner and denser as the radius is increased. The beam propagation properties are controlled by the dimensions of the primary ring radius and width [9, 10] but can also be tailored by introducing a tilt though a linear phase gradient [13, 16] which in the case of these rotationally symmetric beams takes the form of a conical phase distribution.
On the other hand, the case where the secondary rings extend inwards instead of outwards, as shown in Fig. 1(b), is an alternative way to define a rotationally symmetric Airy beam. We call this less studied [14, 17] rotationally symmetric Airy beam as inverse ring-Airy.
The field distribution of both beams can be described as follows:Eq. (1) describes a ring-Airy beam distribution, as shown in Fig. 1(a), while for b = −1 it describes an inverse ring-Airy beam distribution, as shown in Fig. 1(b). Using this unifying description, the primary radius and width, defined as full width at half maximum (FWHM), can be estimated respectively from R0 ≅ r0 + b w and FWHM ≅ 2.28w. Figure 1(c) shows a graphical representation of radial intensity distribution I(r, z = 0) of ring-Airy and inverse ring-Airy beams as the radius parameter is varied. As one can clearly observe, using the description of Eq. (1), there is a smooth transition from the ring-Airy to the inverse ring-Airy beams. Despite the unified analytical description of Eq. (1) for ring and inverse ring-Airy beams, as shown in Fig. 2 their propagation properties are quite different. Ring-Airy beams exhibit abrupt autofocusing keeping very low intensity up to the focus, as shown in Fig. 2 (a), in contrast to inverse ring-Airy beams that exhibit abrupt autodefocus. Furthermore, as shown in Fig. 2(a), 2(b) by varying the conical phase angle we can shift the focus position along the propagation axis. We have to note here that although the typical parabolic trajectory of accelerating ring-Airy beams [9–12], is shadowed by the presence of the linear gradient, the abrupt autofocusing behavior is clearly evident in the profiles of the peak intensity along z.
3. Focal region of rotationally symmetric Airy beams
By analytically describing the focus position and spatial dimensions, one is able to tailor the focal region of any beam. For Gaussian beams this is straightforward and is well described in textbooks . On the other hand, although the recently introduced [9, 10], rotationally symmetric beams have been intensively studied [1, 4, 6, 11, 19, 20], such an analytic description is partially available for ring-Airy beams [11, 13] but, to our knowledge, not for inverse ring-Airy beams. In the following we will provide analytic descriptions of the focal position and spatial dimensions of these rotationally symmetric Airy beams, that enable us to tailor their focal region.
3.1. Colliding Airy beams
An excellent model for the study of ring-Airy beams is their 1D equivalent, the colliding 1D Airy beams [12, 21]. These beams consist of two mirror symmetric 1D Airy beams that propagate along parabolic trajectories. As they propagate these beams coherently interference leading to a focal point along the propagation axis, similar to the autofocus of the ring-Airy beams. Since analytical formulas [15,16] can be used to describe the propagation of an 1D Airy beam we can tailor the focusing behavior of the two colliding 1D Airy beams and expand this to cylindrically symmetric Airy beams. The conical linear phase gradient introduced in Eq. (1) is in this 1D scheme equivalent to an initial tilt angle for each beam. Using the analytical formulas for the propagation of tilted 1D Airy beams  we can describe the field distribution of the colliding 1D Airy beams:Eq. (1) the parameter b can take only two values. For b = 1 we get the 1D equivalent of the autofocusing ring-Airy beam, while for b = −1 we get the 1D equivalent of the abruptly autodefocusing inverse ring-Airy beam. For small values of the truncation coefficient we can predict analytically, the position of the focus: Eq. (3) are a result of the Janus nature of these waves , and correspond to the foci of the "real" and "virtual" waves. Furthermore, by properly manipulating the initial shift and linear gradient we can tailor the distribution of the intensity maximum along propagation. This focus tailoring technique can then be generalized to cylindrically symmetric Airy beams by introducing a conical phase, equivalent to that introduced by an axicon . As shown in Fig. 3 there is a strong analogy between the autofocusing colliding 1D Airy beams Fig. 3(i) and the ring-Airy Fig. 3(ii), as well as between the autodefocusing colliding 1D Airy beams Fig. 3(iv), and the inverse ring-Airy beams Fig. 3(v). Besides the position of the focus, the intensity distribution along the propagation axis, especially in the focal region, is practically identical as shown in Figs. 3(iii), 3(vi). On the other hand, the I(x, z) intensity profiles, as shown in Figs. 3(iii), 3(iv) insets are quite different especially along x direction. This is expected since in one case (colliding Airy) we have the interference of two beams and in other (radially symmetric Airy) the interference of a conical wavefront.
Using the analytic predictions of Eq. (2) that refer to the propagation of colliding 1D Airy beams, and taking into account the above-mentioned similarities and differences, we retrieve approximate analytic expressions for the focal properties of cylindrically symmetric Airy beams.
3.2. Cylindrically symmetric Airy beams
A unique feature of ring-Airy beams is that they abruptly autofocus along propagation, while their peak intensity remains very low up to their abrupt focus. The basic parameters that describe the focus of such a beam are depicted in the insets of Fig. 2(a), 2(b). These include the effective focal length fAi , which denotes the position of the abrupt focus, the focal spot length ΔfAi which denotes the longitudinal (FWHM) of the intensity focal distribution, and focal spot width wAi which denotes the transverse dimensions of the focus (FWHM).
Recently approximate analytic relations  were introduced for the estimation of abruptly autofocusing ring-Airy focal parameters fAi, ΔfAi, wAi as a function of the initial radius r0 and width w parameters, valid only for r0/w ≫ 1. Using the equivalence to 1D+1 Airy beams, we expanded this analysis colliding to cover abruptly autofocusing and autodefocusing ring-Airy beams, under the action of a conical phase gradient. Furthermore, our unified analytical description now covers an extended initial parameters range. In more detail, using the analytic prediction of the trajectory of the peak of the primary intensity lobe of Eq. (3) to radially symmetric Airy beams shaped by a conical phase we get for the focus position:Eq. (3) the Janus nature  of these waves leads to two solutions for the position of the focus. For the case of we get that the two foci are symmetrically positioned relative to z = 0.
The next crucial focal parameter is the focal spot length ΔfAi. Taking into account the trajectory of the 1D toy model of Eq. (3), we can estimate the projection of the primary lobe on the axis at the focus. This enables us to reach to an approximate analytic estimation for the focal spot length:23] where the wavefront cone angle at focus is predicted by the parabolic trajectory of the 1D toy model as described in Eq. (3). Thus, we reach to an approximate analytic estimation of the focal spot width for both ring-Airy and inverse ring-Airy Eq. (6) beams:
3.2.1. Abruptly autofocusing ring-Airy beams
To confirm the validity of our analytic predictions as described in Eqs. (4) – (6) we used numerical simulations of the paraxial wave equation, for a broad range of parameters. More specifically we study the evolution of the normalized focal parameters fAi/zAi, ΔfAi/fAi, wAi/w as a function of two parameters; the normalized shape factor s and the normalized conical angle . In our simulations we either vary the normalized shape factor s keeping a constant angle or we vary keeping s constant. In the first case, since , our results are comparable to the estimations presented in . We have to note though that Eqs. (4) – (6) are valid for a much larger initial parameters range from r0/w > 1, up to r0/w ≈ 40, rad.
Figure 4(a) shows the position of the focus fAi as a function of the primary ring parameters. Numerical simulations are in both cases in excellent agreement with the theoretical predictions of Eq. (4). As the shape factor s is increased, the normalized focal distance is increased proportionally to as predicted by Eq. (4) for . It should be noted here that increasing s by decreasing the ring width w parameter also changes zAi. On the other hand, Fig. 4(b) depicts the effect of the conical phase gradient on the focus position. As expected, increasing the conical angle decreases the normalized focal distance. Clearly, by varying the conical angle , instead of the shape factor s, we achieve a similar dynamic range in controlling the focus position, although fAi saturates for rad. Figure 5 shows the focal spot length ΔfAi as a function of the primary ring parameters. Again the numerical simulation results are in very good agreement to the analytical predictions of Eq. (5). As the shape factor s is increased the normalized focal spot length is decreased. Combining this with the fact that the focal distance fAi is increased, as shown in Fig. 4(a), we get that as the focus is shifted further away it becomes shorter compared to the focal distance fAi. On the other hand, as shown in Fig. 5(b), when the conical angle is increased the the normalized focal length quickly saturates after rad.
Likewise, Fig. 6 shows the dependence of the focal spot width wAi, normalized over the width parameter w as function of the initial parameters. The analytical predictions of Eq. (6) are in very good agreement with the numerical simulation results. As the shape factor s is increased, the normalized focus width monotonically decreases. A similar behavior is observed in Fig. 6(b), where the cone angle is increased.
The shape of the focal region is well described by it’s aspect ratio, defined as the ratio of the focal length to the focal width AR = ΔfAi/wAi. Figure 7 depicts the analytically estimated values of the aspect ratio AR for autofocusing ring-Airy beams as a function of the radius ro and width w parameters. The results in Fig. 7(a) are retrieved for a conical angle θ = 0 while in Fig. 7(b) for a conical angle θ=5 mrad. The contour lines show that a constant aspect ratio, i.e. a preservation of the focus shape, can be achieved by controlling the shape parameter through the radius parameter ro while keeping the width parameter w constant. This interesting behavior was first demonstrated by  for the case of θ = 0. Here we expand this result for the case of a conical angle θ > 0. As it can be seen from Fig. 7 a typical characteristic of these beams is the needle-like shape of the focal region since the aspect ratio is quite high (AR > 100). This from the practical point of view, and in respect of the shape of the focal region, sets autofocusing ring-Airy beams between Bessel beams  where the aspect ratio is much higher and typical Gaussian beams where the aspect ratio can be quite lower.
3.2.2. Abruptly autodefocusing inverse ring-Airy beams
In order to reveal their abrupt autodefocusing behavior inverse ring-Airy beams, and in contrast to autofocusing ring-Airy beams, require the presence of a linear conical phase. This a conical phase gradient is represented by conical angle θ in Eq. (1). As shown in Fig. 2(b) when this conical angle exceeds a value of abrupt autodefocusing behavior is observed. As in the case of autofocusing ring-Airy beams, we used numerical simulations of the paraxial wave equation, to confirm the validity of our analysis for a broad range of parameters, in the regime.
Figure 8(a) shows the position of the focus fAi of an autodefocusing inverse ring-Airy beam as a function of the scale parameter s. The simulations are performed for a conical angle θ = 10 mrad . Numerical simulations are in excellent agreement with the theoretical predictions of Eq. (4). The normalized focus position shows a practically linear dependence on the shape parameter s. Indeed from Eq. (4) and by setting b = −1 we get for . Likewise, Fig. 8(b) shows the effect of the conical phase gradient on the focus position of the abruptly autodefocusing ring-Airy. Again, the numerical simulations are in excellent agreement to the predictions of Eq. (4). The normalized focus position exhibits a monotonic decrease as the conical angle is increased, in similar fashion the the case of the abruptly autofocusing ring-Airy beams shown in Fig. 4. Figure 9(a) depicts the dependence of focal spot length ΔfAi of an autodefocusing inverse ring-Airy beam as a function of the scale parameter s. The numerical simulation results are in excellent agreement to the analytical predictions of Eq. (5). Similarly to the case of the ring-Airy beams shown in Fig. 5(a) the normalized focal length shows a monotonic decrease as the s parameter is increased. On the other hand, as shown in Fig. 9(b), although there is a very good agreement between the numerical simulations and the analytical predictions for rad, Eq. (5) underestimates ΔfAi/fAi by < 5% when rad. This is due to the fact that for low values of the cone angle we are approaching the validity limit of our analysis which in holds for .
Figure 10(a) depicts the focal width wAi of an autodefocusing inverse ring-Airy beam, normalized over the width parameter w, as a function of the shape parameter s. The analytic predictions are again in excellent agreement to the numerical simulations. Although the normalized focal width wAi exhibits a linear dependence on the shape parameter s, this holds only for if the width parameter w remains constant, as we can deduce from Eq. (6) for the case of .
The dependence of wAi/w on the conical phase angle is shown in Fig. 10(b). While the fit between analytical predictions and numerical simulation results is excellent for rad, Eq. (6) overestimates by 10% for rad. As in the case of ΔfAi for such low values we approach validity limit of our analysis. The aspect ratio AR of autodefocusing inverse ring-Airy beams as a function of the radius r o and width parameters w is shown in Fig. 11. The results in Fig. 11(a) are for a conical angle θ = 10 mrad while in Fig. 11(b) for θ = 15 mrad. In both cases the valid AR values are between ro > w and as shown by the respective curves. The contour lines mark the areas of constant aspect ratio. As in the case of abruptly autofocusing ring-Airy beams AR, and thus the focal voxel shape, is preserved when the radius parameter ro is increased while keeping the width parameter w constant. Likewise, the aspect ratio takes large values (AR > 100), indicating a needle-like focal region.
We have studied the properties of the focal region of abruptly autofocusing ring-Airy, and their variant autodefocusing inverse ring-Airy beams, under the action of a conical phase gradient. Exploiting the similarities between the propagation of 1D Airy beams to cylindrically symmetric Airy beams, we present, using a unified formulation for both beam types, analytic approximations for the position, length and width of the focus. Our results enable the tailoring of the focal region of such abruptly autofocusing and autodefocusing beams by simply tuning the initial radius ro and width w parameters. Furthermore, We have confirmed the validity of our analytical results by numerical simulations over a broad parameters range. We expect that our results will have a significant impact to applications that exploit the unique properties of cylindrically symmetric Airy beams like for instance, materials processing, optical trapping, and deposition of high-laser powers at remote locations.
General Secretariat for Research and Technology (GSRT); Hellenic Foundation for Research and Innovation (HFRI) (HRFI-844).
D.M. acknowledges financial support from the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) (HRFI-844).
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